On certain realizations of the q-deformed exterior differential calculus

We investigate two particular realizations of a q-deformed differential calculus at q being a primitive root of unity, qN = 1. Particular attention is paid to the Z3-graded case N = 3. First we construct an analogue of the exterior differential calculus on a manifold, then we introduce a discrete realization of such a calculus on generalized Clifford algebras. Finally, combining both constructions, we discuss a ZN-graded generalization of gauge theory.